[SOUND] So here's a table of values of 
the function F of X equals sin 1 / X. 
F of one is really sin of one, it's like 
8.. 
F of 1. which is really sin of one over 
1. sin of 10 -.5. 
F of 01.. 
This will be sine of 100, it's also about 
-0.5. 
F of 0.001, which is like sine of 1000, 
well that's 0.8 and some more, right? 
So the question is these numbers aren't 
really getting close to anything in 
particular. 
Can you really say that if you evaluate f 
at values which are close to but not 
equal to zero, that the outputs are 
actually getting close to anything in 
particular. 
I mean this is positive, negative, 
negative, positive, negative, positive, 
negative, it's looking pretty bad. 
Instead of a table, let's look at a 
graph. 
Here, I've got a graph of the funtcion f 
(x) = sin 1 / x. 
And you see the middle of this graph is 
just that horrible green blob. 
Right? 
It's really hard to make out any detail. 
You might think that's just a consequence 
of the fact that I'm drawing this graph 
with such thick lines. 
You know, and if I used thinner lines to 
draw my graph, maybe I could, you know 
get rid of this green blob and really see 
some detail. 
Even if I dial down the size of the lines 
that I'm using to draw this graph, the 
blob thing is still there, you know. 
And it's really there in the graph of the 
function. 
Even if these lines were true lines, zero 
thickness, it wouldn't be possible to fit 
even a single atom next to the Y axis 
without touching the graph of this 
function. 
The graph is oscillating wildly near 
zero. 
Even if your input is very close to zero 
your output could be anything between -1 
and 1. 
So in light of this evidence, the limit 
of sine 1 / x as x approaches 0 does not. 
[SOUND] exist. 
Which sometimes I'll abbreviate DNE, for 
does not exist. 
what does it even mean to say it doesn't 
exist? 
What do we mean by the definition of 
limit? 
To say the limit equals something means 
that I can make the output as close as I 
want to l by making x close to a. 
So when I say this limit doesn't exist, I 
mean it's not the case that this limit is 
equal to anything, okay? 
If you tell me this limit is some 
positive number, well look. 
When I evaluate the function at a number 
very close to 0, the output is negative. 
So the limit is probably not some 
positive number but there's also inputs 
very close to zero that give positive 
outputs so that the limit is pulling out 
a negative number either. 
Limit is pulling out zero either cause 
none of these numbers are getting close 
to zero. 
So in this sense. 
This limit just doesn't exist because 
it's not the case that this limit is 
equal to anything in particular. 
If you tell me this limit is equal to l, 
I'm going to show you numbers close to 
zero which aren't close to l. 
Let's see another example along the same 
lines. 
This is a particularly confusing example 
because in the function f(x)x) = sine pi 
/ x. 
The function evaluated at 1 is 0. 
That's pretty clear because that is sine 
of pi and sine of pi is zero. 
About the function at 0.1, I'm counting 
that's also equal to zero. 
The function at 0.01, that is also zero. 
This can be kind of confusing. 
When you take a look here, I typed in sin 
pi divided by 0.01 on to my calculator. 
This is calculating the function's value 
at 01.. 
If I ask my calculator to do this, it is 
not telling me the answer zero, right? 
The calculator's giving me this, 
admittedly, a very small number, right, E 
-11 here. 
But it's still not actually zero. 
So, can I convince you that this is even 
true? 
That the functions value at 01. actually 
is equal to zero. 
What is f (0.01)? 
Well it's the same as sine, of pi / 0.01. 
Now here I'm taking pi and I'm dividing 
it by 0.01. 
That's the same thing as what? 
That's the same thing as multiplying a 
100. 
I'm dividing by a hundredth, that's the 
same as multiplying by a 100. 
So this function at.01 is sine of 100 pi. 
What's sine of 100 pi? 
Well, think back to what the graph of 
sine looks like. 
Here's a graph of sine, zero to two pi. 
If I do it again. 
Here it is at four pi, and I drew it 
again. 
Here it is at six pi, and I'm going to 
keep on going. 
And eventually, I'm going to get to 100 
pi. 
And at that point, sign really is going 
to be equal to zero. 
Calculators are great, they're also 
terrible. 
This calculator can't really calculate 
with pi, all it can do is calculate with 
some approximation to pi. 
We can use our human mind to evaluate 
this function exactly. 
In light of this evidence, you might be 
tricked into believing that the limit of 
f(x) as x approaches 0 is equal to 0. 
After all these points are approaching 
zero, and function evaluating each of 
these points is zero. 
So maybe that means that this is true. 
So it looks like the limit is equal to 
zero. 
But, what happens if I look at some other 
points? 
We'll take a look at this example. 
Here's the same function, f of x equals 
sign of pi over x. 
This function, if I evaluate it at 75. is 
this, maybe a little bit mysterious 
number, negative 866. and so forth. 
If I evaluate this function at 075. you 
get the same thing. 
If I evaluate the function at 0075,. I 
get the same thing. 
At 00075. I get the same thing, .000075, 
I get the same thing. 
So, what's going on here. 
Well what is this number? 
I mean 0.8666. 
This isn't just some sort of random 
number. 
Right? 
This is in fact negative the square root 
of three over two. 
And it looks like this function at all of 
these points has the same value, negative 
the square root of three over two. 
So does that mean that the limit as X 
approaches zero of F of X, [SOUND] is 
equal to negative the square root of 
three over two? 
[SOUND] I mean again all of these values, 
.75.075.0075, these input values are 
approaching zero, and the functions value 
at all of those inputs is the same. 
So, what gives, is the limit zero, is a 
negative point A, which is it?. 
Okay, okay, I've been little bit too 
tricky in picking my input points. 
Since the same function, F of X equals 
Sin pie over X, and here, I am picking a 
collection of points, again approaching 
zero, .7, .07, .007, .0007. 
Its getting closer and closer to zero. 
But now, my output values are looking 
pretty random. 
I mean, they are not over the same, for 
instance. 
So this is, maybe some evidence, that, 
the limit. 
Of sine pi over x as x approaches zero. 
Doesn't exist. 
Here I've got a bunch of input points 
that are getting closer and closer to 
zero but my output values at least don't 
appear to be getting close to. 
We're not just learning. 
We're exploring. 
[SOUND] I encourage you to cook up you 
own examples. 
We've seen a couple examples now of where 
limits don't exist but can you come up 
with more? 
[SOUND]